Could be the grade and a low spot near the foundation. When the sprinkler is on it hits the house and floods the area near the foundation, the water sips into the ground along the foundation and finds a way into the foundation wall cavity and into the basement. Seen that happen on occasion or two.

Check the grade near the foundation and make sure it's not back-pitched, maybe the sprinkler head needs to be adjusted and pointed away from the house. It could be the problem with the line itself, did you sprinkler guy dug around the head to make sure the sprinkler line is not damaged or has a leak in it? There is no water in the sprinkler line when the sprinkler is off. As soon as the control zone valve is open, it releases water pressure and if the sprinkler line is damaged it could leak under the surface and all that water finds a way and sips into the basement.

I would call the sprinkler guy again and have him check everything better and if he has to he should dig around in the area where you have them the problem and expose some of the sprinklers and do a few tests with the sprinkler on.

Good luck